Harmonic balls and the two-phase Schwarz function

Shahgholian, Henrik; Sjödin, Tomas

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Harmonic balls and the two-phase Schwarz function

Shahgholian, Henrik

Royal Institute of Technology, Sweden.

Sjödin, Tomas

Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.

2013 (English)In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 58, no 6, p. 837-852Article in journal (Refereed) Published

Abstract [en]

In this article, we introduce the concept of harmonic balls in sub-domains of ⁿ, through a mean-value property for a subclass of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept fails to exist due to the fact that analytic functions cannot have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exist, and gives rise to the generalization of the well-known Schwarz function to the case of a two-phase Schwarz function. Our primary goal is to derive simple properties for these problems, and tease the appetites of experts working on Schwarz function and related topics. Hopefully these two concepts will provoke further study of the topic.

Place, publisher, year, edition, pages

Taylor and Francis: STM, Behavioural Science and Public Health Titles / Taylor and Francis , 2013. Vol. 58, no 6, p. 837-852

Keywords [en]

Schwarz function, mean-value property, harmonic functions, two-phase free boundary, quadrature domains, Primary: 35R35, 31A05, 31B05, 31B20

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Natural Sciences

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URN: urn:nbn:se:liu:diva-94602DOI: 10.1080/17476933.2011.622046ISI: 000319380200010OAI: oai:DiVA.org:liu-94602DiVA, id: diva2:633607

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Funding Agencies|Swedish Research Council|

Available from: 2013-06-27 Created: 2013-06-27 Last updated: 2017-12-06

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